Question:
The area bounded by the curves \(y = |x|-1\) and \(y = -|x|+1\) is
The area bounded by the curves \(y = |x|-1\) and \(y = -|x|+1\) is
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The two V-shaped graphs form a square; integrate the gap between them.
Updated On: Oct 1, 2026
- \(1\) sq.unit
- \(2\) sq. units
- \(2\sqrt{2}\) sq.units
- \(4\) sq.units
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
\(y = 1 - |x|\) is an upward-opening-down tent with peak at \((0,1)\). \(y = |x| - 1\) is a V with its vertex at \((0,-1)\). They meet where \(|x| - 1 = 1 - |x|\), that is \(|x| = 1\).
Step 2: Set up the integral:
The region lies between \(x = -1\) and \(x = 1\), with the tent above the V. By symmetry:
\[ A = 2\int_0^1\left[(1 - x) - (x - 1)\right]dx = 2\int_0^1 (2 - 2x)\,dx \]
Step 3: Evaluate:
\[ 2\left[2x - x^2\right]_0^1 = 2(2 - 1) = 2 \]
Step 4: Check by geometry:
The four vertices are \((0,1), (1,0), (0,-1), (-1,0)\), a square with diagonals of length 2. Its area is \(\frac12 \times 2 \times 2 = 2\).
Step 5: Why the other options are wrong.
Area 1 is half of the square, 4 is the area of a square with diagonal 2\(\sqrt2\), and \(2\sqrt2\) would be its perimeter-style value, not the area.
Final Answer:
The area is 2 sq. units, option (B). \[ \boxed{2\text{ sq. units}} \]
\(y = 1 - |x|\) is an upward-opening-down tent with peak at \((0,1)\). \(y = |x| - 1\) is a V with its vertex at \((0,-1)\). They meet where \(|x| - 1 = 1 - |x|\), that is \(|x| = 1\).
Step 2: Set up the integral:
The region lies between \(x = -1\) and \(x = 1\), with the tent above the V. By symmetry:
\[ A = 2\int_0^1\left[(1 - x) - (x - 1)\right]dx = 2\int_0^1 (2 - 2x)\,dx \]
Step 3: Evaluate:
\[ 2\left[2x - x^2\right]_0^1 = 2(2 - 1) = 2 \]
Step 4: Check by geometry:
The four vertices are \((0,1), (1,0), (0,-1), (-1,0)\), a square with diagonals of length 2. Its area is \(\frac12 \times 2 \times 2 = 2\).
Step 5: Why the other options are wrong.
Area 1 is half of the square, 4 is the area of a square with diagonal 2\(\sqrt2\), and \(2\sqrt2\) would be its perimeter-style value, not the area.
Final Answer:
The area is 2 sq. units, option (B). \[ \boxed{2\text{ sq. units}} \]
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